PARAMETRIC EQUATIONS OF NONHOLONOMIC NONCONSERVATIVE SYSTEMS IN THE EVENT SPACE AND THE METHOD OF THEIR INTEGRATION

In this paper,the parametric equations with multipliers of nonholonomic nonconservative sys- tems in the event space are established,their properties are studied,and their explicit formulation is obtained. And then the field method for integrating these equations is given.Finally,an example illustrating the appli- cation of the integration method is given.

Parametric Equation of Stress Concentration Factor for Circular X-Joints Under Axial Loads

In engineering practice, tubular X-joints have been widely used in offshore structures. The fatigue failure of tubular X-joints in offshore engineering is mainly caused by axial tensile stress. In this study, the stress concentration factor distribution along the weld toe in the hot spot stress region for tubular X-joints subject to axial loads have been analyzed by use of finite element method. Through numerical analysis, it has been found that the peak stress concentration factor is located at the saddle position. Thereafter, 80 models have been analyzed, and the effect of the geometric parameters of a tubular X-joint on the stress concentration factor has been investigated. Based on the experimental values of the numerical stress concentration factor, a parametric equation to calculate the stress concentration factor of tubular X-joints has been proposed. The accuracy of this equation has been verified against the requirement of the Fatigue Guidance Review Panel, and the proposed equation is found capable of producing reasonably accurate stress concentration factor values for tubular X-joints subject to axial loads.

SOLUTION TO A PARAMETRIC EQUATION WITH GENERALIZED BOUNDARY CONDITION IN TRANSPORT THEORY

This paper deals with the solution of a parametric equation with generalized boundary condiiton in transport theory. It gives the distribution of parameter (so called delta-eigenvalue [1]) with which the equation has non-zero solution. A necessary and sufficient condition for the existence of; he control critical eigenvalue delta0 is established.

Meta-Auto-Decoder:a Meta-Learning-Based Reduced Order Model for Solving Parametric Partial Differential Equations

Many important problems in science and engineering require solving the so-called parametric partial differential equations(PDEs),i.e.,PDEs with different physical parameters,boundary conditions,shapes of computational domains,etc.Typical reduced order modeling techniques accelerate the solution of the parametric PDEs by projecting them onto a linear trial manifold constructed in the ofline stage.These methods often need a predefined mesh as well as a series of precomputed solution snapshots,and may struggle to balance between the efficiency and accuracy due to the limitation of the linear ansatz.Utilizing the nonlinear representation of neural networks(NNs),we propose the Meta-Auto-Decoder(MAD)to construct a nonlinear trial manifold,whose best possible performance is measured theoretically by the decoder width.Based on the meta-learning concept,the trial manifold can be learned in a mesh-free and unsupervised way during the pre-training stage.Fast adaptation to new(possibly heterogeneous)PDE parameters is enabled by searching on this trial manifold,and optionally fine-tuning the trial manifold at the same time.Extensive numerical experiments show that the MAD method exhibits a faster convergence speed without losing the accuracy than other deep learning-based methods.

THE PROPER PARAMETRIZATION OF A SPECIAL CLASS OF RATIONAL PARAMETRIC EQUATIONS

We give a proper reparametrization theorem for a set of rational parametric equations which is proper for all but one of its parameters. We also give an algorithm to determine whether a set of rational parametric equations belongs to this class, and if it does, we reparametrize it such that the new parametric equations are proper.

PARAMETRIC EQUATION SOLVING AND QUANTIFIER ELIMINATION IN FINITE FIELDS WITH THE CHARACTERISTIC SET METHOD

For a parametric algebraic system in finite fields, this paper presents a method for computing the cover and the refined cover based on the characteristic set method. From the cover, the author knows for what parametric values the system has solutions and at the same time presents the solutions in the form of proper chains. By the refined cover, the author gives a complete classification of the number of solutions for this system, that is, the author divides the parameter space into several disjoint components, and on every component the system has a fix number of solutions. Moreover, the author develops a method of quantifier elimination for first order formulas in finite fields.

IMPLICITIZATION OF PARTIAL DIFFERENTIAL RATIONAL PARAMETRIC EQUATIONS

In this paper, we propose algorithms for the following problems in the implicitization of a set of partial differential rational parametric equations P. （1）To find a characteristic set for the implicit prime ideal of P; （2） To find a canonical representation for the image of P; （3）To decide whether the parameters of P are independent, and if not, to re-parameterize P so that the new parametric equations have independent parameters; （4） To compute the inversion maps of P, and as a consequence, to decide whether P is proper.

Parametric equations for notch stress concentration factors of rib—deck welds under bending loading

The effective notch stress approach for evaluating the fatigue strength of rib-deck welds requires notch stress concentration factors obtained from complex finite element analysis.To improve the efficiency of the approach,the notch stress concentration factors for three typical fatigue-cracking modes(i.e.,root-toe,root-deck,and toe-deck cracking modes)were thoroughly investigated in this study.First,we developed a model for investigating the effective notch stress in rib-deck welds.Then,we performed a parametric analysis to investigate the effects of multiple geometric parameters of a rib-deck weld on the notch stress concentration factors.On this basis,the multiple linear stepwise regression analysis was performed to obtain the optimal regression functions for predicting the notch stress concentration factors.Finally,we employed the proposed formulas in a case study.The notch stress concentration factors estimated from the developed formulas show agree well with the finite element analysis results.The results of the case study demonstrate the feasibility and reliability of the proposed formulas.It also shows that the fatigue design curve of FAT225 seems to be conservative for evaluating the fatigue strength of rib-deck welds.

SOLUTION TO A PARAMETRIC EQUATION IN TRANSPORT THEORY

In 1976, Ronen et al. raised the question of how to solve the new integrodifferential parameter equation with transport theory. So far there have only been some approximate calculations and numerical analyses about this question. Using functional analysis, this note discusses this question in a strict mathematical way, gives the parameter distribution in L^p space (1≤p≤+∞) with which the

Using Parametric Mathematical Modeling to Develop a Geometric and Topological Intuition for Molecular Knots

Knot theory is a branch of topology in pure mathematics, however, it has been increasingly used in different sciences such as chemistry. Mathematically, a knot is a subset of three-dimensional space which is homeomorphic to a circle and it is only defined in a closed loop. In chemistry, knots have been applied to synthetic molecular design. Mathematics and chemistry together can work to determine, characterize and create knots which help to understand different molecular designs and then forecast their physical features. In this study, we provide an introduction to the knot theory and its topological concepts, and then we extend it to the context of chemistry. We present parametric representations for several synthetic knots. The main goal of this paper is to develop a geometric and topological intuition for molecular knots using parametric equations. Since parameterizations are non-unique;there is more than one set of parametric equations to specify the same molecular knots. This parametric representation can be used easily to express geometrically molecular knots and would be helpful to find out more complicated molecular models.

On the Family of Thue Equation |x^3+mx^2y-(m+3)xy^2+y^3|=k

The family of cubic Thue equation which depend on two parameters ｜ x^3 ＋ mx^2 y-（m＋3） xy^2＋y^3｜=k is studied. Using rational approximation, we give a smaller upper bound of the solution of the equation, that is quite better than the present result. Moreover, we study two inequalities ｜ x^3 ＋ mx^2y-（m ＋ 3） xy^2＋y^3 ｜ =k≤2m＋3 and ｜x^3 ＋mx^2y- （m＋3）xy^2 ＋ y^3｜ = k≤ （2m＋3）^2 separately. Our result of upper bound make it easy to solve those inequalities by simple method of continuous fraction expansion.

THE APPLICATION OF THE VARIATIONAL PRINCIPLE IN THE CONSTRAINED CONTROL SYSTEM

The regular equations on the constraint variables are established for LQ and non-linear control problems in this paper,then the extreme-value principles of con-straint variables are discussed for the equality and unequality constraint cases respec-tively. At last, the given example verifies the conclusion of this paper.The work in this paper will lay the foundation for the further study about the constrained LQ and non-linear control systems.

Optimal analytical and numerical approximations to the(un)forced(un)damped parametric pendulum oscillator

The(un)forced(un)damped parametric pendulum oscillator(PPO)is analyzed analytically and numerically using some simple,effective,and more accurate techniques.In the first technique,the ansatz method is employed for analyzing the unforced damped PPO and for deriving some optimal and accurate analytical approximations in the form of angular Mathieu functions.In the second approach,some approximations to(un)forced damped PPO are obtained in the form of trigonometric functions using the ansatz method.In the third approach,He’s frequency-amplitude principle is applied for deriving some approximations to the(un)damped PPO.In the forth approach,He’s homotopy technique is employed for analyzing the forced(un)damped PPO numerically.In the fifth approach,the p-solution Method,which is constructed based on Krylov–Bogoliúbov Mitropolsky method,is introduced for deriving an approximation to the forced damped PPO.In the final approach,the hybrid Padé-finite difference method is carried out for analyzing the damped PPO numerically.All proposed techniques are compared to the fourth-order Runge–Kutta(RK4)numerical solution.Moreover,the global maximum residual distance error is estimated for checking the accuracy of the obtained approximations.The proposed methodologies and approximations can help many researchers in studying and investigating several nonlinear phenomena related to the oscillations that can arise in various branches of science,e.g.waves and oscillations in plasma physics.

LUROTH'S THEOREM IN DIFFERENTIAL FIELDS

In this paper, we present a constructive proof of Luroth's theorem in differential case. We also give a method to find the inversion maps for general differential rational parametric equations. As a consequence, we prove that a differential rational curve always has a set of proper parametric equations.

Static behavior of offshore two-planar tubular KT-joints under axial loading at fire-induced elevated temperatures

A total of 540 nonlinear steady-state finite element analyses were performed to study the influence of temperature and dimensionless geometrical parameters(β,γ,θ,andτ)on the ultimate strength,failure modes,and initial stiffness of two-planar tubular KT-joints.The joints were analyzed under two types of axial loading and five different temperatures(20℃,200℃,400℃,550℃,and 700℃).So far,there has not been any equation available for calculating the ultimate strength of two-planar tubular KT-joints at elevated temperatures.Hence,after parametric study,a set of design formulas were developed through nonlinear regression analyses,to calculate the ultimate strength of two-planar tubular KT-joints subjected to axial loading at elevated temperatures.

Output Feedback Stabilization of Spacecraft Autonomous Rendezvous Subject to Actuator Saturation

This paper studies the output feedback dynamic gain scheduled control for stabilizing a spacecraft rendezvous system subject to actuator saturation. By using the parametric Lyapunov equation and the gain scheduling technique, a new observer-based output feedback controller is proposed to solve the semi-global stabilization problem for spacecraft rendezvous system with actuator saturation. By scheduling the design parameter online, the convergence rates of the closed-loop system are improved. Numerical simulations show the effectiveness of the proposed approaches.

Robust Global Stabilization of Spacecraft Rendezvous System via Gain Scheduling

The problem of robust global stabilization of a spacecraft circular orbit rendezvous system with input saturation and inputadditive uncertainties is studied in this paper. The relative models with saturation nonlinearity are established based on ClohesseyWiltshire equation. Considering the advantages of the recently developed parametric Lyapunov equation-based low gain feedback design method and an existing high gain scheduling technique, a new robust gain scheduling controller is proposed to solve the robust global stabilization problem. To apply the proposed gain scheduling approaches, only a scalar nonlinear equation is required to be solved.Different from the controller design, simulations have been carried out directly on the nonlinear model of the spacecraft rendezvous operation instead of a linearized one. The effectiveness of the proposed approach is shown.

REDUCED BASIS METHOD FOR PARAMETRIZED ELLIPTIC ADVECTION-REACTION PROBLEMS

In this work we consider the Reduced Basis method for the solution of parametrized advection-reaction partial differential equations. For the generation of the basis we adopt a stabilized finite element method and we define the Reduced Basis method in the ＂primal- dual＂ formulation for this stabilized problem. We provide a priori Reduced Basis error estimates and we discuss the effects of the finite element approximation on the Reduced Basis error. We propose an adaptive algorithm, based on the a posteriori Reduced Basis error estimate, for the selection of the sample sets upon which the basis are built; the idea leading this algorithm is the minimization of the computational costs associated with the solution of the Reduced Basis problem. Numerical tests demonstrate the efficiency, in terms of computational costs, of the ＂primal-dual＂ Reduced Basis approach with respect to an ＂only primal＂ one. Parametrized advection-reaction partial differential equations, Reduced Basis method, ＂primal-dual＂ reduced basis approach, Stabilized finite element method, a posteriori error estimation.

Smooth Time-Varying Feedback Control of Nonholonomic Systems with Applications to the Attitude Control of Underactuated Spacecraft

This paper studies the control of chained nonholonomic systems by using smooth time-varying feedback.First,by using a time-varying state transformation,the chained nonholonomic system is transformed into a linear time-varying system.Second,with the aid of some properties of a class of parametric Lyapunov equations and time-varying systems theory,smooth time-varying state feedback controller is developed to ensure the exponential convergence of both the system states and controls.Different from existing results,the design parameters in the proposed control strategy are independent of the initial conditions of the system.Third,as an application of the proposed smooth time-varying feedback,the attitude control problem for underactuated axisymmetric spacecraft is addressed.Finally,a numerical simulation is presented to validate the effectiveness of the proposed method.